Some numerical and experimental advances in chaotic scattering - - PowerPoint PPT Presentation

Some numerical and experimental advances in chaotic scattering Microlocal Analysis and Spectral Theory 2013

Maciej Zworski

UC Berkeley

September 28, 2013

A scattering problem

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V (x) =

3

• j=1

aje−|x−xj|2/bj We consider ih∂tu = −h2∆u + V (x)u

ih∂tu = −h2∆u + V (x)u

Newtonian dynamics: x′(t) = 2ξ(t), ξ′(t) = −∇V (x(t)), ϕt(x(0), ξ(0)) := (x(t), ξ(t)). Trapped set at energy E: KE := {(x, ξ) : ξ2 + V (x) = E, ϕt(x, ξ) → ∞, t → ±∞}.

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In the movies we saw the effects of Newtonian (classical) dynamics but we also saw oscillations, concentration and decay of waves. Quantum Resonances describe these waves resonating in interaction regions: there exist complex numbers zj(h) = Ej(h) − iΓj(h), Γj(h) > 0, and wj(x) ∈ L2 (resonant states), such that (P − zj(h))wj = 0, wj is outgoing .

Quantum Resonances describe the resonating waves:

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Computed using squarepot.m http://www.cims.nyu.edu/∼dbindel/resonant1d/

Here is how they sound: time = linspace(0,500,5000); sound(real(exp(-i*z*time)))

A real experimental example Potzuweit–Weich–Barkhofen–Kuhl–St¨

• ckmann–Z ’12

Resonances for three discs: Barkhofen–Kuhl–Weich ’13

Resonances for three discs:

2

1

O3 O2

1

s s2 s3 ν

1

x ϕ ξ ν

O

Barkhofen–Kuhl–Weich ’13

incoming set trapped set

• utgoing set

Poon–Campos–Ott–Grebogi ’96

Resonances for three discs: Resonant states are microlocalized on the outgoing set: Helffer–Sj¨

• strand ’85, Bony–Michel ’04, Nonnenmacher–Rubin ’07.

Sj¨

• strand ’90:

Suppose P = −h2∆ + V where V is analytic (and reasonable). Suppose that the classical flow is hyperbolic on KE. Then resonances of of P, zj(h), satisfy #{zj(h) ∈ [E−ǫ, E+ǫ]−i[0, h]} ≤ Ch−m/2, m > dim ∪|E ′−E|<2ǫKE ′. Here the dimension is the Minkowski/box dimension: for M ⊂ Rk, codim M = sup{γ : lim sup

ǫ→0

ǫ−γvolRk({ρ : d(ρ, M) < ǫ}) < ∞}. Earlier, non-geometric bounds: Regge ’58, Melrose ’82, Intissar ’86, Z ’87,’89.

Sj¨

• strand ’90:

#{zj(h) ∈ [E−ǫ, E+ǫ]−i[0, h]} ≤ Ch−m/2, m > dim ∪|E ′−E|<2ǫKE ′. KE := {(x, ξ) : ξ2 + V (x) = E, ϕt(x, ξ) → ∞, t → ±∞}. codim M = sup{γ : lim sup

ǫ→0

ǫ−γvolRk({ρ : d(ρ, M) < ǫ}) < ∞}.

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More recently: Sj¨

• strand–Z ’07:

Resonances for −h2∆ + V where V ∈ C∞

c (Rn; R) (and more

general operators) #{zj(h) ∈ [E − h, E + h] − i[0, h]} ≤ Ch−µ, 2µ + 1 > dim KE. Nonnenmacher–Sj¨

• strand–Z ’13:

Resonances for −∆ on Rn \ J

j=1 Oj (and more general operators). !(")

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Numerical studies: Lin ’02:

Lu–Sridhar–Z ’03: The reason for showing the paper is to indicate that to communicate an idea it helps to publish it in physics.

Dyatlov ’13 (math) , Dyatlov–Z ’13 (physics) Weyl law for quasi-normal modes/resonances for perturbations of Kerr-de Sitter metrics (rotating black holes).

Dyatlov ’13 (math), Dyatlov–Z ’13 (physics) Weyl law for quasi-normal modes/resonances for perturbations of Kerr-de Sitter metrics (rotating black holes). The trapped set as a changes from 0 to 1: Wunsch–Z ’11: The key property of this smooth trapped set is the r-normal hyperbolicity for any r. Hirsch–Pugh–Schub ’77: stable under small C r perturbations.

Dyatlov ’13 (math), Dyatlov–Z ’13 (physics) Weyl law for quasi-normal modes/resonances for perturbations of Kerr-de Sitter metrics (rotating black holes). When the transversal expansion rates satisfy νmax < 2νmin (valid for 98% of rotation speeds of black holes) then #{zj ∈ the blue box} = λ2 (2π)2 vol(∪E<1KE)(1 + o(1)), Sj¨

• strand–Z ’99: Asymptotics for resonances for convex obstacles

satisfying a pinching condition (cubic bands).

Dyatlov ’13 (math), Dyatlov–Z ’13 (physics) Weyl law for quasi-normal modes/resonances for perturbations of Kerr-de Sitter metrics (rotating black holes). When the transversal expansion rates satisfy νmax < 2νmin (valid for 98% speeds of rotation of the black hole) then #{zj ∈ the blue box} = λ2 (2π)2 vol(∪E<1KE)(1 + o(1)), Faure–Tsujii ’13: Similar asymptotics for the Policott–Ruelle resonances for contact Anosov flows.

Dyatlov ’13 (math), Dyatlov–Z ’13 (physics) Weyl law for quasi-normal modes/resonances for perturbations of Kerr-de Sitter metrics (rotating black holes). When the transversal expansion rates satisfy νmax < 2νmin (valid for 98% speeds of rotation of the black hole) then #{zj ∈ the blue box} = λ2 (2π)2 vol(∪E<1KE)(1 + o(1)), Faure–Tsujii ’13: Similar asymptotics for the Policott–Ruelle resonances for contact Anosov flows.

A simpler model. Nonnenmacher–Z ’05, ’07’: quantized open Baker maps (Balazs–Voros ’89, Saraceno ’90) Classical relation: (q, p) ∼ (q′, p′) ⇐ ⇒

• q′ = 3q,

p′ = p/3, 0 ≤ q ≤ 1/3 q′ = 3q − 2, p′ = (p + 2)/3, 2/3 ≤ q < 1. Quantum operator: MN = F∗

3N

  FN FN   . (FP is the discrete Fourier transform on CP).

Open Baker map: incoming set trapped set

• utgoing set

Three discs reduced to the boundary:

Open Baker map: Expected fractal Weyl law: for 0 < r < r0 < 1, ♯{λ ∈ Spec(MN), |λ| > r} ∼ N

log 2 log 3 ,

MN = F∗

3N

  FN FN   .

Nonnenmacher–Z ’07: for a simplified quantum Baker map corresponding to a complicated classical chaotic relation we have the fractal Weyl law for a sequence N = 3k (the Walsh model).

Recent works in physics using variants of the quantum open maps (and other methods): Schomerus–Tworzyd lo ’04, Keating et al ’06, Wiersig–Main ’08, Ramilowski et al ’09, Pedrosa et al ’09, Shepelyansky ’09, Shomerus–Wiersig–Main ’09, Ermann–Shepelyansky ’10, Kopp–Schomerus ’10, Ebersp¨ acher–Main–Wunner ’10, K¨

• rber et al

’13. An interdisciplinary example:

A yet different setting: manifolds with hyperbolic ends Resonances defined as poles of (−∆X − (n − 1 − s)s)−1, continued from Im s > (n − 1)/2; X is a manifold with hyperbolic ends. Fractal upper bounds: Z ’99: Γ\H2, Γ convex co-compact (based on Sj¨

• strand ’90)

Lin–Guillop´ e–Z ’04: Γ\H2, Γ a Schottky group (based on some new Selberg zeta function techniques) Datchev–Dyatlov ’13: any manifold with hyperbolic ends (based on Sj¨

• strand-Z ’07 and a new approach to meromorphic continuation

by Vasy ’13) Other models using zeta functions: hyperbolic rational maps. Here the growth of zeros of the zeta function is related to the dimension

• f the Julia set. Strain-Z ’03, Christianson ’05.

Borthwick ’13:

Borthwick ’13: ℓ1 = 10, ℓ2 = 12, ϕ = 2π/5

Borthwick ’13 Comparison with the fractal Weyl law:

Potzuweit–Weich–Barkhofen–Kuhl–St¨

• ckmann–Z ’12

Experimental investigation of fractal Weyl laws.

Potzuweit–Weich–Barkhofen–Kuhl–St¨

• ckmann–Z ’12

Experimental investigation of fractal Weyl laws. Left: The counting functions for R/a = 2, 2.25, 3.9 Fits of their slope for high frequencies are shown in blue. The orange curve

• ver the lower histogram corresponds to the Weyl formula with

12% loss. Plotted in the inset is the difference between the Weyl formula with 12% loss and the experimental counting function for the closed system (R/a = 2).

Potzuweit–Weich–Barkhofen–Kuhl–St¨

• ckmann–Z ’12

Experimental investigation of fractal Weyl laws. Right: The data points correspond to the fitted exponent of the counting function in dependence of the R/a parameter. The three squares mark the examples which have already been presented in the previous figures. The darker shaded blue region indicates the R/a values without open channels; lighter shaded blue region has

• nly a few open channels.

This may not seem to be so succesful but it lead to an interesting experiment about the gap between the real axis and resonances. Barkhofen–Weich–Potzuweit–Kuhl–St¨

• ckmann–Z ’13

We look for γ > 0 such that there are no resonances in Im z > −γ, Re z > C0

How do we determine that gap at the high frequency limit when the dynamics is hyperbolic? Gaspard-Rice ’89, Lu-Sridhar-Z ’03, Barkhofen et al ’12 Ikawa ’88, Burq ’93, Nonnenmacher-Z ’09, Naud ’04,’12, Petkov-Stoyanov ’11

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We define the topological pressure associated to the unstable Jacobian: J+

t (ρ) = det

• dΦt

|E +

ρ

• PE(s) = lim

T→∞

1 T log

• T−1<Tγ<T

J+(γ)−s , where γ are closed orbits with period Tγ. Ikawa ’88, Nonnemacher-Z ’09, Petkov-Stoyanov ’11: There are no resonances with Im λ > PE(1/2) (at high energies)

There are no resonances with Im λ > PE(1/2) (at high energies) The decay of correlations is closely related to resonance free strips. Potzuweit-Weich-Barkhofen-Kuhl-St¨

• ckmann-Z, PRL ’13

Lu-Sridhar-Z ’03: concentration of decay rates at P(1)/2, PRL ’03

It is also seen in the case of scattering on hyperbolic sufaces. Borthwick ’13: Naud ’13: If dim K1 = 2δ + 1 then #{sj : σ < Resj, | Im sj| < r} = O(r1+τ(σ)), where τ(σ) < δ for σ < δ/2. Fractal Weyl law (Z ’99, Lin-Guillop´ e-Z ’04, Datchev-Dyatlov ’13) gives the bound r1+δ for all σ.

Thank you!